hep-th/0107259
SCIPP-01/27
arXiv:hep-th/0107259v1 30 Jul 2001
Dark Matter and Dark Energy: A Physicist’s
Perspective1
Michael Dine
Santa Cruz Institute for Particle Physics, Santa Cruz CA 95064
Abstract
For physicists, recent developments in astrophysics and cosmology present exciting challenges. We are conducting “experiments” in energy regimes some of which will be probed
by accelerators in the near future, and others which are inevitably the subject of more speculative theoretical investigations. Dark matter is an area where we have hope of making
discoveries both with accelerator experiments and dedicated searches. Inflation and dark
energy lie in regimes where presently our only hope for a fundamental understanding lies in
string theory.
1
Physics summary talk of the conference The Dark Universe: Matter, Energy and Gravity, Space Telescope
Institute, April 2001
1
Introduction
It is a truism that the development of astronomy, astrophysics, cosmology relies on our understanding of the relevant laws of physics. It is thus no surprise that my astronomy colleagues
tend to know more classical mechanics, electricity and magnetism, atomic and nuclear physics
than my colleagues in particle theory.
As we consider many of the questions which we now face in cosmology, we must confront the
fact that we simply do not know the relevant laws of nature. The public often asks us “what
came before the big bang.” We usually think of this as requiring understanding of physics
at the Planck scale. But at present we can’t even come close. Ignorance sets in slightly
above nucleosynthesis, and becomes severe by the time we reach the weak scale. Some of the
questions which trouble us will be settled by experiment over the next decades; some require
new theoretical developments. Needless to say, it is possible that much will remain obscure for
a long time.
• GeV scales: QCD is by now a well tested theory, but the phase structure of QCD is
not completely understood, and possible first order phase transitions, superconducting
phases, strange matter, etc. could be relevant both to astrophysics and cosmology. These
questions may be settled by improved lattice gauge calculations, and conceivably by developments at RHIC.
• T = 100 GeV-TeV: This is the regime of the weak phase transition. In order to understand
this transition, we need experimental information on the Higgs particle, or whatever
physics is responsible for the mass of the W and Z. This physics might be the origin of
the matter-antimatter asymmetry[1]. This is physics which will be explored by the Large
Hadron Collider at CERN and by a large electron-positron collider, hopefully to be built
by an international consortium over the next decade.
• T = 100 Gev-TeV: One of the best-motivated candidates for the dark matter is the
“LSP” (Lightest Supersymmetric Particle) expected if nature is supersymmetric. If the
supersymmetry hypothesis is correct, we can expect to encounter this particle and its
supersymmetric cousins at the Tevatron or LHC and the electron-positron collider. This
physics could well be responsible for the matter-antimatter asymmetry.
• T = 1 Tev: There have been several suggestions over the past three years that the
fundamental scale of physics might lie at the TeV scale[2, 3, 4]. In this case, the Planck
2
scale would be so large – and gravity so weak – because there are some very large or
highly warped extra dimensions of space. If this hypothesis is correct, there could well be
dramatic new phenomena in cosmology just above the temperature of nucleosynthesis.
• T = 1010 Gev? The axion is another well-motivated dark matter candidate. It is associated with a symmetry known as a Peccei-Quinn symmetry. 1010 GeV might be the scale
at which this symmetry is broken. It also might be a scale associated with the dynamics
responsible for supersymmetry breaking.
• T = 1015 GeV? This could well be the scale of the physics responsible for inflation. Over
the past few years, this scale has also emerged as a possible value for the fundamental
scale of M theory[5]. The two might well be connected[6].
• T > 1015 GeV? Perhaps physics at these scales holds the explanation of the value of the
cosmological constant, and identification of the nature of the dark energy. Perhaps only
here lies the physics which resolves the singularity of the big bang, and explains the initial
conditions of the universe.
For particle physicists it is extremely exciting to think that there are
connections between events in accelerators and our understanding of the history of the
universe. Perhaps as important, cosmology can serve as a testing grounds for ideas which are
not so readily studied in more conventional experiments. We are probably not going to answer
all of the questions which I have listed here soon. But it is remarkable that, as we will see, we
have hopes of attacking all of them.
2
String or M Theory
I will take string theory as a theoretical umbrella in this talk. String theory is a natural
framework to talk about all of the issues I have raised above. Indeed, for many of these
questions, it is the only framework we have. First, string theory is our only consistent theory of
gravity and quantum mechanics. Such a framework is essential if we are to address many of the
questions which we face in cosmology. Equally important, string theory encompasses virtually
every idea we have for dark matter and energy:
• Low energy supersymmetry, with symmetries like R parity which give rise to a stable,
weakly interacting particle.
3
• Axions: As I will explain further below, string theory is the only theoretical context in
which we can make sense of the axion hypothesis.
• Cosmological constant: String theory is the only theoretical framework in which we can,
even in principle, calculate the cosmological constant. It has realizations all of the various
proposed solutions: it has candidates for multiple vacua which might produce an anthropic
solution, as mentioned in Vilenkin’s talk; it can produce extremely light particles, which
realize the other anthropic proposal which Vilenkin mentioned[7]; and it is “holographic”
(to be explained below), so it might offer entirely
new solutions.
• Quintessence: String theory is the only context in which we can sensibly discuss the sorts
of extremely flat potentials necessary to realize the ideas of quintessence[8],[9].
What is also striking about string theory is that it will allow us to make rather definite
statements about many of these ideas. We will see that within our current understanding of
string theory, the two anthropic solutions which I mentioned above are implausible.1 Physicists tend to view anthropic explanations of features of physical law with skepticism or worse.
Personally, I have for many years thought we might have to contemplate an anthropic solution
of the cosmological constant problem[10, 11, 12]. I realized at this meeting that astronomers
are more receptive to these ideas than physicists. But what is significant here is that we can
potentially use string theory to rule out some anthropic explanations on scientific rather than
philosophical grounds.
While I will not stress the point here, by similar reasoning, the idea of quintessence similarly
extremely difficult to realize in string theory[9].
2.1
String Theory: A Quick Introduction and Survey of Recent Developments
What is string theory? At the most simple level, it is just that: a theory of quantized strings.
Such a theory is automatically a theory which is generally covariant with non-abelian gauge
groups. Why? While there has been much progress in understanding these theories, we have
at best only a glimpse as to the answer to this question.
1
This statement requires some qualification; see [37].
4
More generally, string theory is a framework in which we might hope to address a variety
of questions both in particle physics and cosmology. While it is often said to be a theory in ten
dimensions, it has solutions with different numbers of dimensions, including four, and
• Standard model gauge interactions
• Repetitive generations (e.g. 3) of quarks and leptons
• Low energy supersymmetry
• Discrete symmetries (R-parity)
• Axions
• Light scalars with very flat potentials (inflatons? quintessence?)
• Exotic possibilities, such as large “compact” dimensions, with dramatic possible implications for particle physics, astrophysics and cosmology.
In the last few years, there have been a number of developments, which are usually grouped
together under the heading of duality:
• What were once thought to be several independent string theories have been recognized to
be states of one large theory (sometimes called M theory)[13]. Given that the difficulties
of quantizing gravity seem so immense, the fact that all previously successful attempts
are part of one structure suggests that, just as there is a unique theory of fundamental
vector bosons interacting with matter, so there may truly be a unique theory of gravity.
• Many interesting dualities have been understood. For example, many string (and field)
theories exhibit an exact electric-magnetic duality (see Jackson, chapter 6!)
• Many new theoretical tools have been developed, which have permitted the study of
quantum aspects of black holes and other real phenomena of quantum gravity. For example, the Beckenstein-Hawking entropy has been understood through the counting of
microscopic states[14].
• A striking new principle of quantum gravity has been discovered, known as the Holographic principle[15]: quantum theories of gravity have far fewer degrees of freedom than
conventional quantum field theories, such as those of the Standard Model. The number grows like the surface area of the system rather than the volume. This is likely to
5
have profound consequences for the understanding of the question of the cosmological
constant[16, 17, 18] and other issues in cosmology[19].
These developments have provided a number of new insights into longstanding problems.
For example, we used to think that the basic scales in string theory would be of order the
Planck scale. But with the new developments, we have recognized new possibilities.
• In the strong coupling limit, string theory is best described in terms of an eleven dimensional theory, with gravity propagating in all eleven dimensions, while gauge interactions
are confined to ten dimensional walls[5]. There is some evidence that this limit is the best
suited for describing the real world[20]. If this idea is correct, the fundamental scale of
this theory, the eleven dimensional Planck mass, satisfies:
M11 ∼ 1015 GeV
(1)
The eleventh dimension is curled up, along with the other (more conventional(!?)) six,
with radii R11 and R given roughly by:
R11 M11 ∼ 10 − 30
6
R6 M11
∼ 60
(2)
The values of GN , and the unification of the gauge couplings, give support for this picture.
• Traditionally, in thinking about compactification, one imagined that any extra dimensions
were extremely small, of order the Planck mass or unification scale. In recent years, it has
been appreciated that extra dimensions might be far larger[3], or could be highly curved[4].
Either possibility, it has been suggested, might provide an alternative to supersymmetry
as a solution to the hierarchy problem. (Prior to these developments, while large extra
dimensions had occasionally been suggested, it had not been possible to make sense of
them.) These new proposals involve walls or branes in a crucial way, much as in the eleven
dimensional limit. The fundamental scale of physics lies at 1 TeV, or so; the smallness of
Newton’s constant is due to the large size of the extra dimensions, through the relation:
GN = Mf8und Vcomp
Mf und ≈ TeV.
(3)
The eleven dimensional picture predicts low energy supersymmetry, but also possesses
scales of the sort needed to understand the features of inflation[6]. The large dimension idea
predicts:
6
• Dramatic growth of cross sections for production of Kaluza-Klein modes (e.g. in the
process e+ + e− → γ + missing energy).
• Cosmology: effects of Kaluza-Klein modes might be important just above nucleosynthesis.
For example, some dimensions might be much smaller at early times.
2.2
What Makes String Theory Hard?
What makes string theory hard? Why don’t we have all the answers? Part of the answer is
simply that it is an ambitious theory. It’s supposed to explain all the facts of the standard
model, with no parameters. It is not reasonable to expect all of the answers to fall out so easily.
But there are also some specific problems:
• While I said that there are states with desirable properties, there are in some sense too
many states. For example, there are states with 11,10,9,8,...4,3 dimensions; states with
or without supersymmetry; states with 1-100’s of generations, and so on.
• The classical solutions possess continuous parameters. From the perspective of “low energy” physicists, these are associated with fields. These fields are called moduli. Examples
include a field called the dilaton, whose expectation value determines the values of the
gauge couplings;, and the radius. We will denote these by g−2 (xµ ) (“dilaton”) and R2 (xµ ),
respectively.
• Cosmology: Moduli are candidates for the inflaton, but they also lead to a set of cosmological difficulties.
• Cosmological constant (more later)
3
DARK MATTER
Particle physics has provided at least two plausible candidates for dark matter.
• The lightest supersymmetric particle (LSP): requiring that the proton lifetime be long in
supersymmetric theories almost inevitably means that the LSP is stable. Supposing that
supersymmetry is broken at a scale of order 1 TeV, automatically leads to a relic density
for this particle of roughly the right order to be the dark matter[21].
7
In supersymmetric field theories, it is necessary to postulate discrete symmetries from
nowhere to explain the stability of the proton; in string theory, such symmetries are
ubiquitous[22]. We heard at this meeting descriptions of ongoing searches for these particles. While the hints from the DARMA experiment are controversial, the 2.6σ discrepancy
in (g − 2)µ provides some cause for optimism that direct evidence for supersymmetry will
soon be found[23]. Indeed, a number of physicists have argued for some time that if the
supersymmetry hypothesis is correct, one is likely to see a discrepancy in g − 2[24]. Over
the next year, further data will be analyzed and the error bars will shrink significantly.
• Axions: The axion is associated with strong CP problem[25]. The axion idea predates
the realization that string theory possesses axions by several years[26], but it is in string
theory that the idea finds a natural home, and indeed it would inevitably would have
been discovered there had it not been suggested earlier. In field theory, the Peccei-Quinn
solution of the strong CP problem requires that one postulate that nature has a symmetry,
which is broken only by tiny quantum effects in the strong interactions. This symmetry
must hold so accurately that extremely tiny gravitational effects would spoil it, and it has
sometimes been argued that this is implausible[27]. But in string theory, Peccei-Quinn
symmetries of exactly the desired type automatically arise. Prior to the understanding
of duality, the Peccei-Quinn scale in string theory was most naturally identified as Mp ,
so if the axions constituted the dark matter, they were undetectable. With the new
understanding, many other possibilities have emerged[28].
4
The Problem of the Dark Energy
As we have heard at this meeting, the evidence for dark energy is mounting. As Professor Livio
stressed in his summary, a year ago many astronomers would have doubted the existence of dark
energy. Now most, if not totally convinced, are starting to believe it. As Professor Perlmutter
remarked, it is particle physicists, especially theorists, who have been his biggest skeptics. As
we will see, this is because the result is so surprising. But given that it now seems likely that
the data – and its interpretation as dark energy – are correct, it is necessarily a profound clue
to the nature of physics at some very different scale.
¿From the perspective of a particle theorist, the question is: what is the energy density of
the vacuum, i.e. of the ground state of whatever is the underlying theory of nature. Obviously
it is a tall order to compute this – we need to know the theory – but dimensional analysis
8
suggests we are in trouble. Particle physicists like to describe the cosmological constant, Λ, as
a quantity of dimensions of [mass]4 . So
Λ = M4
(4)
where M should be some characteristic mass scale in physics. Is M = Mp (the Planck mass)?
M = MZ ? M = mp (the proton mass)? Even in the last case, we would be off by 47 orders of
magnitude! Could there be some principle which simply predicts zero? If so, what is the origin
of the very tiny observed value?
In string theory, there is some good news with regards to this problem. At the classical
level, all of the string vacua I have mentioned have vanishing cosmological constant. While
technically easy to describe[22], this fact is in many ways mysterious. It is not a consequence
of symmetries of space time. So this fact represents a striking failure of dimensional analysis,
of just the sort we want!
However, even if Λ = 0 at the classical level of some theory, it is very hard to understand
why quantum effects wouldn’t generate a huge value for it. The quantum theories of the standard model describe approximately free fields, i.e they are collections of harmonic oscillators,
one for each momentum, spin (and other quantum numbers). The ground state energy of such
a theory is then, to lowest order in h̄,
Λ=
X
(−1)F
Z
d3 k ~ 2
k + m2 .
(2π 3 )
q
(5)
In general, this expression is very divergent. In our understanding of effective quantum field
theories, so successful in describing the standard model, this means that Λ is just a parameter
of the theory. This is why physicists (with a few exceptions) traditionally ignored this problem.
On the other hand, at some level, if this is the correct way to think about the vacuum
energy, some physics must cut off this integral. What might this be? If nature were exactly
supersymmetric, then the bosonic and fermionic contributions to Λ would cancel. If nature is
approximately supersymmetric, the integral diverges quadratically, and assuming that physics
at the Planck scale provides the cutoff,
2
Λ ∼ Msusy
Mp2 ≈ (1010 GeV)4 ?
(6)
In string theory, there are no divergences. Since this is a theory of gravity, the calculation
of the cosmological constant should be well defined. So this should be a good test of string
theory. Does it pass? Is Λ = 0? 10−47
9
Here we have the not so good news: We don’t know the answer to this question. All of the
string vacua we understand possess moduli at the classical level. These are the light fields with
no potential which I referred to above. The expectation values of these fields determine the
coupling constant of the string theory, and the masses of various states. Often we can compute
a potential for these moduli, but the potential always tends to zero as the coupling tends to
zero. Indeed, we have no examples where we can find a stable minimum of the potential in
a completely controlled approximation, since, almost by definition, our approximations break
down at such a point.
When one discusses moduli, with potentials which fall to zero at infinity, it is natural to
consider quintessence. So far, I have spoken of the dark energy as a cosmological constant. As
we have heard, many authors have considered the possibility that the dark energy represents
some form of quintessence, which I will loosely refer to as the energy of some time-varying
field. This is an interesting idea, if only as the equation of state for such a field provides a
measure of the quality of future experiments to study the dark energy. It should be noted that,
whatever the details of the underlying theory, the mass of the quintessence field today (the
second derivative of its potential) can not be significantly smaller than the current horizon size.
This is an extremely small number in particle physics units. In other words, not only must the
actual value of the present energy density be extremely tiny but so must other quantities.
In string theory, however, it is hard to make sense of the quintessence idea, precisely for
the reasons I gave above. The difficulty is that, in examples we can analyze, the scale of the
potential is connected to the scale of supersymmetry breaking, which is much too large. So in
some sense, one needs to fine tune not only the scale of the potential, but also its derivatives,
with extreme precision[9]. (Since this
talk was presented, two papers have appeared noting that there are also serious conceptual
issues with quintessence in string theory[29, 30].) Quintessence also does not provide a simple
explanation of the “why now” puzzle: the question why the cosmological constant, now, is
comparable to the energy density of dark matter. For example, in a simple model[31], this
question is resolved by fine-tuning an additional parameter, at the percent or fraction of a
percent level. This is not to say that observers should not focus their efforts on measuring w.
At the very least, such measurements will give us further confidence that there is a large dark
energy component.
Given that quintessence does not fit easily into our current understanding of string theory,
let us return to the more conventional cosmological constant idea. If there are stable minima,
10
they are in regions where we can’t calculate. Do such stable states exist? Does the cosmological
constant vanish, or is it very small for such states (as a consequence of some principle)? Might
there be many such states so that we could implement an anthropic solution?
Here there is (reasonably) good news:
• String theory possesses features which allow us to discuss anthropic solutions of the cosmological constant of the type discussed by Vilenkin at this meeting. It has been argued
that there might be a very
dense set of vacuum states[32, 33]. There can be very flat potentials.
• String theory is not like field theory. There is good evidence that it does not possess
nearly as many degrees of freedom as field theory. So perhaps the naive quantum estimate
we described above is not correct. We might then hope that the classical cosmological
constant vanishes, and that the quantum contributions are much smaller than naively
expected.
It is interesting that both of theses ideas suggest that the cosmological constant is very
small, but not zero.
5
The Anthropic Principle
To the “why now” question, a number of answers have been offered. Through the years, many
authors have noted that an energy scale of order 1TeV is a natural scale to consider in physics,
and that G2N [TeV]8 is within an order of magnitude of the observed dark energy (in the past, it
was argued that it was within such a factor of the limit on the cosmological constant). This is
quite impressive, until one remembers that we are indeed trying to explain a coincidence within
a factor of two, and that the choice of T eV is very rough. E.g. if it happened that the correct
scale was 3 TeV, we would be off by nearly 104 !
Unfortunately, one can’t help but look at the data and conclude that it is pointing us in
the direction of some sort of anthropic explanation. Perhaps, if the cosmological constant were
much different than observed, the conditions for life, even in its most rudimentary conceivable
form, might not be satisfied? I observed at this meeting that astronomers are less afraid to
contemplate such a prospect than physicists; they are aware of numerous coincidences in nature
which may require such an explanation. In the company of many of my physics colleagues,
11
mentioning the cosmological constant is viewed as barely better than advocating creationism.
As Weinberg has remarked: A physicist talking about the anthropic principle runs the same
risk as a cleric talking about pornography: no matter how much you say you’re against it, some
people will think you’re a little too interested.
This topic has been reviewed by Vilenkin at this meeting, and I will only add a few remarks.
What I have in mind by the anthropic principle in this context is what Weinberg calls the “Weak
anthropic principle.” The idea is that the universe is vastly larger than what we see within
our horizon. In different regions of the universe, the cosmological constant, and possibly other
physical constants, take different values. Then just as people can only live on planets with water,
atmospheres, etc. (or, just as fish can only live in water), galaxies/stars/planets/people can only
exist in a tiny fraction of the full universe. From galaxy formation, it was originally argued that
this hypothesis could not explain a cosmological constant as small as observed[10, 34]. More
refined arguments give results which may be compatible with what we see[7].
As Vilenkin described, there have been a variety of proposals as to how the laws of nature
might admit such variation of the parameters. One possibility is that the system has a huge
(discrete) number of possible (metastable) ground states, and the distribution of the corresponding energies is nearly continuous[36, 32, 33]. This quasicontinuous distribution of states
has been dubbed a “discretum”[32]. Note that the number of states must be enormous. If, for
example, the typical scale of the energies is of order 1 TeV, then the number of states must be
at least
of order 1061 . A second possibility is that the universe is permeated by an extremely light
field, with Compton wavelength large compared to the present horizon. As a result this field
is currently frozen, but during the inflationary era, it fluctuated over a range of values, large
enough that in some regions it cancelled any preexisting cosmological constant.
One may not find this mode of explanation appealing, but in some sense it may not matter.
Within string theory, one can argue that neither of these proposed explanations is very plausible.
Consider, first, the possibility of a very light field, φ. The mass of this field has to be smaller
than 10−50 GeV. There are mechanisms in string theory which could produce a particle this
light. But these mechanisms all imply that the maximal value of the field is of order Mp . But
in order to cancel off a cosmological constant of order 1012 GeV 4 , we need φ ∼ 1040 Mp or so[35]!
In [36], it was argued that a peculiar type of axion, known as the “irrational axion,” might
give rise to a suitable discretum, but subsequent searches have failed to turn up any examples of
12
the required phenomenon in string theory. Four form fluxes in string theory might provide the
necessary “discretum” to understand the cosmological constant[32, 33]. Whether this works
in detail requires resolving many difficult questions. For example, we need to understand
the stability not of just one but of 10120 (or so)states. It also raises the specter that all
quantities (the gauge coupling constants, the masses of the elementary particles....) would all
be determined anthropically (or alternatively would be random numbers)[37]. It is hard to
imagine that all of the standard model parameters are anthropic. Nor do they look like random
numbers. So while this idea is the most difficult of the set to rule out, it does not see particularly
promising.
In sum, the remarkable coincidence of the cosmological constant and the present dark
matter density is very suggestive of an anthropic explanation. But an anthropic explanation,
to be scientific, requires a sensible underlying theory, presumably in the context of a theory
which is capable of making other predictions. So far, we don’t have such a theory.
6
The Holographic Principle
’t Hooft and Susskind argued, from considerations of black hole physics, that in a sensible theory
of gravity, in a region of volume V and surface area A, the number of degrees of freedom must be
proportional to A[15]. The most familiar piece of evidence for this is the Beckenstein-Hawking
entropy formula:
S=
GN A
.
4
(7)
Other features of black hole physics also support this. The fact that in some sense the information about what is going on in a large volume is encoded in degrees of freedom residing on
the surface is the origin of the term holographic.
There is some evidence that string theory is holographic:
• Naive notions about numbers of degrees of freedom are not correct in string theory. For
example, string ground states in smaller dimensions of space time have more degrees of
freedom (suggesting that compactified theories are more “fundamental” than uncompactified ones).
• String perturbation theory has holographic features: the S matrix seems to be the crucial
observable; C. Thorn argued long ago that the perturbation theory itself has one the
13
degrees of freedom of a theory in d − 1 dimensions.
• Two non-perturbative formulations of string theory are known (the “Matrix Model” and
the AdS-CFT correspondence). Both are explicitly holographic.
What might be the implications of this principle for the Cosmological Constant? These are
not clear, but they seem likely to be dramatic, since in the cosmological constant expression,
eqn. 5, one might
no longer have V
R
d3 k, but instead a sum over far fewer degrees of freedom. Are there few
enough? The problem is not sufficiently well understood to say at the present time[17].
This sort of reasoning has lead to even more radical conjectures. In string theory, there
seem to be states with varying numbers of dimensions, and varying amounts of supersymmetry.
Many states with unbroken supersymmetry can be argued to be exact solutions of the theory.
Susskind has suggested that perhaps De Sitter space is not allowed in string theory. He offers
no solid argument, but points to hints based on holography. Banks proposes that we think very
differently about the question of supersymmetry breaking[16]. He argues that the number of
states in De Sitter space is finite. Given that recent observations suggest that our universe is De
Sitter. What determines the number of states? Banks proposes that this number is a parameter.
The cosmological constant and the amount of supersymmetry breaking are determined by this
parameter! This proposal explains why states with too much supersymmetry might not be
viable (and it is the only proposal which does so). It requires that in holographic theories a
different relation between Λ and the scale of supersymmetry breaking holds. There is some
reason to believe this might be the case. It requires, however, a very surprising relation to hold
between the cosmological constant and the scale of supersymmetry breaking.
Both ideas are highly speculative, and they are not (yet) supported by a substantial amount
of evidence. But they are suggestive. Indeed, if nothing else, they indicate the sorts of radical
rethinking of many of our basic ideas in physics which may be required to understand the dark
energy.
7
Conclusions
Particle physicists are eager to know the answers to the questions:
• What is the dark matter?
14
• Is there really dark energy? What is its equation of state?
My experimental colleagues are very interested in dark matter searches, SNAP and ground
based proposals to study Type Ia Supernovae. Theorists are hopeful that they have predicted
the correct form of the dark matter; they are frantically trying to explain the dark energy. Both
are sure to lead to important insights into fundamental law.
7.1
Acknowledgments
I would like to thank many of my colleagues for discussions, and for educating me about the
issues raised here, especially Tom Banks, Nathan Seiberg and Leonard Susskind. I would like
to thank the organizers and participants in this conference for a very stimulating experience,
and particularly to thank Mario Livio. This work was supported in part by the U.S. Department
of energy.
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precursors of these ideas can be found in [2].
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[5] P. Horava and E. Witten, “Eleven-Dimensional Supergravity on a Manifold With Boundary,” Nucl.Phys. B475 (1996) 94, hep-th/9603142; “Heterotic and Type I String Dynamics
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15
[6] T. Banks, “Remarks on M Theoretic Cosmology,” hep-th/9906126.
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